Approximate functional inequalities by additive mappings (Q2914890)

Let \(f:G \to Y\), where \(G\) is an \(n\)-divisible abelian group and \(Y\) is a Banach space. After showing that \(f\) is additive if and only if it satisfies the inequality NEWLINE\[NEWLINE \left\| f(x)+f(y)+nf(z) \right\| \leq \left\| nf\Big(\frac{x+y}{n}+z\Big) \right\|, \quad x,y,z \in G, NEWLINE\]NEWLINE the authors prove the following stability result:NEWLINENEWLINETheorem: Suppose that \(f:G \to Y\) satisfies the inequality NEWLINE\[NEWLINE \left\| f(x)+f(y)+nf(z) \right\| \leq \left\| nf\Big(\frac{x+y}{n}+z\Big) \right\|+\phi(x,y,z) NEWLINE\]NEWLINE where \(\phi: G^3 \to [0,\infty)\) is a given function such that NEWLINE\[NEWLINE \lim_{k \to \infty} \frac{1}{n^k}\phi(n^kx,n^ky,n^kz)=0 NEWLINE\]NEWLINE for all \(x, y, z \in G\) and NEWLINE\[NEWLINE \check{\phi}(x,z):=\sum_{i=0}^{\infty} \frac{1}{2n^{i+1}} \Big(\phi(n^{i+1}x,0,-n^iz)+\phi(-n^{i+1}x,0,n^iz)\Big)<\infty NEWLINE\]NEWLINE for all \(x, z \in G\). Then there exists a unique additive mapping \(h:G \to Y\) such that NEWLINE\[NEWLINE \left\| f(x)-h(x)\right\| \leq \check{\phi}(x,x)+\frac{\phi(x,-x,0)}{2}+\frac{n^2}{n-1}\left\| f(0)\right\| NEWLINE\]NEWLINE for all \(x \in G\).NEWLINENEWLINEMoreover, the paper contains corollaries and other similar results.